HAL Id: hal-02987523
https://hal.inria.fr/hal-02987523
Submitted on 3 Nov 2020
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Shape Representations and Evolution Schemes
Marc Schoenauer
To cite this version:
Marc Schoenauer. Shape Representations and Evolution Schemes. Fifth Annual Conference on Evo- lutionary Programming, Sep 1996, San Diego, United States. �hal-02987523�
Shape Representations and Evolution Schemes
Marc Schoenauer
CMAP { URA CNRS 756, Ecole Polytechnique
91128 PalaiseauCedex, France
Marc.Schoenauer@polytechn ique .fr
Abstract
The choice of a representation i.e. the deni- tion of the search space, is of vital importance in all Evolutionary Optimization processes. In the context of Topological Optimum Design in Structural Mechanics, this paper investigates pos- sible representations for evolutionary shape de- sign. The goal is the identication of a shape in IRn (n= 2 orn = 3) having optimal mechanical properties. Evolutionary Computation has been demonstrated a valuable tool for TOD problems.
However, all past results are based on the straight- forward bitstring representation whose complex- ity increases with that of the underlying Mechan- ical model. To overcome this diculty, dier- ent representations for shapes are introduced, and compared on the benchmark problem of TOD us- ing various evolutionary schemes. The results are discussed with respect to the dierent degrees of epistasis of the representations.
1 Introduction
Representation is long acknowledged a central issue for Evolutionary Computation. From early discus- sions on the now out-of-date [Michalewicz 1992] prob- lem of encoding real parameters in GAs (e.g. binary vs Gray coding [Caruna and Schaer 1988]) to recent works on comparing representations for the TSP problem [Radclie and Surry 1994], EC researchers have tried to characterize the desirable properties of the mapping from the genotype space, where evolution takes place, into the phenotype space, where environmental pres- sure acts [Fogel 1995a, Fogel 1995b]. However, most of these works either present general recommendations and heuristics (e.g. the degree of degeneracy in the represen- tation should be as small as possible to avoid loosing in- formation) or focus on xed-length representations (e.g.
binary encoding for the TSP problem).
This paper studies variable-length representations for two- or three-dimensional shapes, in the eld of Topological Optimum Design (TOD): the goal is to nd a structure (a shape in a given design do- main) having prescribed mechanical properties and
minimal weight. The straightforward representa- tion for shapes that has been exclusively used in past evolutionary attempts on Optimum Design prob- lems [Jensen 1992, Chapman, Saitou and Jakiela 1994, Chapman and Jakiela 1995,
Kane, Jouve and Schoenauer 1995] is a bitstring repre- sentation based on a mesh of the design domain. How- ever, this representation hardly scales up, neither when considering 3-D shapes, nor when the accuracy of the mechanical behavior of the structure (depending on the size of the mesh) becomes a central issue.
Therefore two other representations are designed; both allow to dissociate the representation of the shape and the accuracy of the tness computation. These represen- tations are variable-length representation with a high de- gree of degeneracy; nevertheless preliminary results show they signicantly outperform the (non-degenerate) bit- array representation.
Forthcoming section 2 presents the Mechanical back- ground of the Optimum Design problem and briey re- calls the results obtained by standard deterministic ap- proaches.
Section 3 describes how Evolutionary Computation addresses some limitations of these standard methods, by handling 2-D shapes as bitstrings (rather standing for bit-arrays). Specic evolution operators have been designed to overcome the geometrical bias of standard operators. Nevertheless this representation is intrinsi- cally limited in the sense that the accuracy of the tness is linked to the size of the chromosome.
Section 4 therefore introduces two new representations for 2-D and 3-D shapes, overcoming the above limitation.
These are experimentally studied on a benchmark prob- lem of TOD, and compared using the three main evolu- tionary schemes, namely GAs, EP and ES. Our results suggest the performance depends on two main features:
the degree of epistasis of a representation, i.e. the degree the expression of one gene depends on the other genes, and the symmetry of the representation with respect to that of the optimal solutions.
Last, some avenues for further research are presented.
1
2 Topological Optimum Design
2.1 Background
Optimum Design in Structural Mechanics consists in nding the best design for a structure in an initial do- main of IR2 or IR3.
More precisely, identifying a shape amounts to nd a partition of the design domain (a subset of IRn, n = 2 orn= 3), into two subsets; one subset is the structure while the other subset represents void1.
The optimality criterion is determined by the mechan- ical properties of the structure, depending on the consti- tutive law (behavior) of the material; only hyper-elastic materials [Ciarlet 1978] will be considered in the follow- ing.In most cases of Optimum Design, the goal is to min- imize the weight of the structure (i.e. the amount of material), while meeting some engineering requirements for given loading cases (e.g. forces, pressure, prescribed displacements,:::). This problem is of utter importance for part manufacturers, as a small decrease in the weight of some widely used part results in large cuts in the man- ufacturing cost.
2.2 Deterministic state of the art
Two contexts are distinguished in Structural Optimum Design:
When the solution is sought as the continuous defor- mation of a given initial shape, iterated small modi- cations of the shape are a deterministic way toward the solution. The methods of domain variation, or sensitivity analysis[Cea 1981], are based on gradient- like optimization techniques in that context.
But continuous deformations do not aect the topol- ogyof the structure (i.e. its number of holes).
When the topology of the solution is unknown, the problem amounts to Topological Optimum Design (TOD); the only deterministic method in that con- text, to the best of the author`s knowledge, is the homogenization method [Bendsoe and Kikushi 1988, Allaire and Kohn 1993]. Homogenization proceeds by rst relaxing the problem and considering proba- bilistic shapes in the design domain: the density of material ranges in [0;1] instead of being either 0 (for void regions) or 1 (for the structure itself). Theoret- ical results ensure that the optimal solution lies in this superset of probabilistic shapes, and can be ap- proximated by deterministic gradient-based methods.
1A formally equivalent problem is that of inclusion identication [Constantinescu 1994], where the goal consists of identifying the repartition of two materials with dierent mechanical properties from the global behavior of the material.
Ω M
Figure 1: A TOD benchmark problem: The 21 can- tilever plate. The design domain is xed on its left bound- ary and a force is applied at point M.
The optimal probabilistic shape is then mapped into a real shape. Spectacular results in two and three di- mensions have been obtained by the homogenization method [Allaire & al. 1996]. The greatest limitation of this approach is that dealing with probabilistic shapes is possible so far in the frame of linear elas- ticity only. Moreover, even in this restricted frame, homogenization method can handle neither multiple loading cases, nor loading applied on the unknown boundary (e.g. a uniform pressure on one side of the structure).
2.3 A TOD benchmark
The classical benchmark problem of topological opti- mization in two dimensions is the cantilever plate, de- scribed in Figure 1: a plate is xed on one of its bound- ary, and one (or more) punctual forces are applied on the other end of the design domain. The goal is to minimize the weight of the structure while staying below some up- per bounds on the maximal displacement of the points where the forces are applied.
A shape is therefore evaluated from two criterions: its weight (or equivalently its area in the two-dimensional case), and its mechanical behavior under the prescribed loading. The numerical simulation of the mechanical be- havior of the structure is achieved using the Finite El- ement Method (FEM) [Zienkiewicz 1977, Ciarlet 1988], which implies a discretization of the shape into small elements, termed mesh.
Formally, the TOD problem is a constrained problem:
one aims at minimizing the weight of the shape, while satisfying the prescribed mechanical requirements.
3 \Standard" Evolutionary TOD
This section briey describes to-date results obtained on TOD problems in the frame of Evolutionary Computa- tion.
3.1 Bitarray Representation
As stated above, the computation of the tness of a shape involves meshing that shape. In that context, the most natural representation for shapes is the bit-array repre- sentation, directly based on a xed regular mesh of the design domain into rectangular elements. Each element of the mesh is given a boolean value (0 or 1) indicat- ing which subset it belongs to (material of void). This representation is straightforward from the point of view of both FEM (this is the simplest possible mesh of the domain) and GAs (the representation can be viewed as a bitstring). All previous stochastic works addressing the topological optimum design problem have adopted that representation, be they based on simulated an- nealing [Ghaddar, Y. Maday, and A. T. Patera 1996] or GAs [Jensen 1992, Chapman, Saitou and Jakiela 1994, Chapman and Jakiela 1995].
3.2 Evolution Operators
However, standard genetic operators appear ill-suited for the above representation of shapes, in the sense they suf- fer from geometrical bias. Bit-arrays are not bit-string:
one-dimensional operators can only exchange horizontal parts of the design domain; if the good schemata involve vertical bands of the structure, these tend to be disrupted by any one-dimensional crossover, and the building block hypothesis therefore does not apply.
Two-dimensional crossover operators, exchanging two- dimensional regions of the shapes (e.g. rectangular blocks or regions separated by random lines), have there- fore been purposely designed. It has been demonstrated on the cantilever plate problem that these specic two- dimensional crossover operators clearly outperform the standard one- and two-points bitstring crossover opera- tors [Kane and Schoenauer 1995].
3.3 Constrained Optimization
Many methods have been designed to constrained evolu- tionary optimization (see [Michalewicz 1995] for a survey of such methods). In particular, the method described in [[Schoenauer and Xanthakis 1993] has been successfully applied to another problem of structural mechanics, the optimization of truss structures. But as the focus of this work is representation, the standard method of penal- ization was chosen, being a non specic robust way of handling constraints.
The rst draft for the tness function thus has the following expression:
F=Area+(DMax DLim)+ (1) whereDMaxis the maximal displacement of the struc- ture when the prescribed force is applied (computed us- ing the FEM),DLimthe imposed limit value for the dis-
placement andis a positive user-supplied penalty pa- rameter (a+ denotes the positive part of a). Though some mechanical diculties in fact lead to slightly more complex expression of the tness (e.g. struc- tures not connecting the xed boundary and the loading are not valid solutions, see [Kane and Schoenauer 1995, Schoenauer 1995]), equation (1) will be considered in the following for the sake of simplicity.
3.4 Successes
Using specic two-dimensional evolution operators on bitarray representation, EC has successfully tack- led TOD problems that could not be addressed by other techniques [Kane, Jouve and Schoenauer 1995, Kane 1996] | at the expense of large computational time.
The most signicant results are the following:
Overall, EC can accommodate
any mechanical model
for which there exists a numerical simulation algorithm. This is conrmed as the results obtained for the large displacement model appear to be the rst results in Optimum Design of structures in non- linear elasticity.The optimization can take into account more than one loading case (as in the design of a bicycle), as well as loading applied on the unknown boundary of the structure (as in the case of the underwater dome).
In some situations, many optimal, or near-optimal
solutions exist. Evolution-
ary algorithms, using for instance the sharing scheme [Goldberg and Richardson 1987], are able to provide the engineer with a range of such solutions, allowing him to take into account inarticulate criterions.
3.5 Drawbacks
The limitations of these results come from the following fact.
The accuracy of Evolutionary TOD is dictated by the size of the mesh underlying the FEM analysis and the tness computation: the above mentioned results were obtained on rather coarse meshes (e.g. 1020) whereas real-world problems and accurate analyses of the mechanical behavior of the shapes require much ner meshes (e.g. 100200).
Increasing the size of the mesh would not only increase the cost of the tness computation (which is roughly quadratic in the size of the mesh) | but also the size of the chromosomes. And increasing the size of the indi- viduals would require to increase in turn the size of the population and the number of generations to reach the same level of convergence (Cerf [Cerf 1994, Cerf 1996]
proved that the minimal size of the population for a GA
to converge increases linearly in term of the size of the bitstring).
Finally the bit-array representation poorly scales up when rening the mesh | not to speak of handling 3-D shapes...
Therefore other representations have been designed in order to overcome this limitation, and dissociate the complexity of the representation and the accuracy of the evaluation.
4 Variable-length representations for shapes
4.1 The Vorono representation
A possible way of representing shapes comes from com- putational geometry, more precisely from the Vorono
diagram theory. The ideas of Vorono diagrams are al- ready well-known in the FEM community, as a powerful tool to generate good meshes [George 1991]. However, the representation of shapes by Vorono diagrams and their evolutionary optimization seems to be original.
4.1.1 Vorono diagrams
Consider a nite number of points V0;:::;VN (the Vorono sites) of a given subset of IRn (the design do- main). To each siteVi is associated the set of all points of the design domain for which the closest Vorono site isVi, termed Vorono cell. The Vorono diagram is the partition of the design domain dened by the Vorono
cells. Each cell is a polyhedral subset of the design do- main, and any partition of a domain of IRn into poly- hedral subsets is the Vorono diagram of at least one set of Vorono sites (see [Preparata and Shamos 1985, Boissonnat and M. Yvinec 1995] for a detailed introduc- tion to Vorono diagrams, and a general presentation of algorithmic geometry).
Consider now a (variable length) list of Vorono sites, each site being labeled 0 or 1. The corresponding Vorono
diagram represents a shape (a partition of the design do- main into two subsets), if each Vorono cell is labeled as the associated site (here the Vorono diagram is sup- posed regular, i.e. to each cell corresponds exactly one site). Example of Voronorepresentations can be seen in Figure 2. The Vorono sites are the dots in the center of the cells. Note that Vorono representation of shapes does not depend in any way on the mesh that will be used to compute the behavior of the shapes. Further- more, Voronodiagrams being dened in any dimension, the extension of this representation to IR3 and IRn is straightforward.
Parent 1 Parent 2
Offspring 1 Offspring 2 Figure 2: The Voronorepresentation crossover operator.
A random line is drawn across both diagrams, and the sites on one side are exchanged
4.1.2 Evolution operators
The evolution operators on the Vorono representation are inspired from both the two-dimensional crossover op- erators designed for the bit-array representation and the usual operators for variable length representations:
The crossover operators exchange Vorono sites on the basis of geometrically-based choice. In this re- spect it is similar to the specic bitarray crossover described in [Kane and Schoenauer 1995]; moreover, this mechanism easily extends to any dimension [Kahng and Moon 1995]. Figure 2 demonstrates an application of this crossover operator.
a rst mutation operator performs a Gaussian muta- tion on the coordinates of the sites, or randomly ips the boolean attribute of some sites;
a "standard" mutation for variable-length represen- tations adds or deletes some sites from the list.
An important remark is that this representation presents a high degree of epistasis (the inuence of one site on the physical shape is modulated by all neighbor sites). This will be discussed in more details in section 5.1.
Practically, the tness of all shapes is evaluated using the same xed mesh2. A shape described by Voronosites is thus mapped on this xed mesh: the subset (material
2This intends to limit the bias due to the unavoidable numerical noise of FEM (the ner the mesh, the lower the numerical error,
or void) an element belongs to is determined from the label of the Voronocell in which the center of gravity of that element lies.
4.2 H-representation
Another representation for shapes is based on an old- time heuristic method in TOD: from the initial design domain, remove material where the mechanical stress is minimal, until the constraints are violated. However, the lack of backtracking makes this method useless in most TOD problems. Nevertheless, this idea gave birth to the
\holes" representation [Dejonghe 1993], later termed H- representation.
4.2.1 The representation
The design domain is by default made of material, and a (variable length) list of \holes" describes the topology of the structure. These holes are elementary shapes taken from a library of possible simples shapes. Only rectangu- lar holes are considered at the moment. On-going work [Seguin 1995] is concerned with other elementary holes (e.g. triangles, circles).
Example of structures described in the H-repre- sentation are presented in Figure 3. The rectangles are taken in a domain larger than the design domain, in or- der not to bias the boundary parts of the design domain.
4.2.2 Evolution operators
The evolution operators are quite similar to those of the Vorono representation:
crossover by geometrical (2D or 3D) exchange of holes (see Figure 3 for an example);
mutation by Gaussian modication of the character- istics (coordinates of the center, width and length) of some holes;
mutation by addition or deletion of a hole;
The H-representation, as the Vorono representation, is independent from any mesh, and hence its complex- ity does not depend on any required accuracy for the simulation of the underlying physical phenomenon. Its merits and limitations will be discussed in the light of the experimental results presented in next section.
As for the Vorono representation, the simulated be- havior of the shapes is computed on a given xed mesh, to limit the numerical noise due to re-meshing. The cri- terion to decide which subset an element does belong to, the higher the computational cost). Hence, the tnesses of dierent structures that are to be compared should be performed with the same mesh. Otherwise, numerical noise due to re meshing might hide the actual dierences in the mechanical behavior of dierent structures.
Figure 3: The H-representation crossover operator. A random line is drawn across both structures, and the holes on one side are exchanged.
is based on whether its center of gravity belongs to a hole (in which case the whole element is void) or not.
4.3 Preliminary comparative results
The same benchmark problem of Optimal Design has been used for the rst comparative results for the three representations of shapes presented in preceding section (see [Schoenauer 1995]). As expected, and even for a rather coarse mesh, the bit-array representation is out- performed by all other representations in terms of com- putational cost as well as in terms of quality of the so- lution. Hence, it will not be considered any more in the rest of the paper.
Both the Vorono representation and the H- representation derive a quasi-optimal solution. But these solutions are more rapidly found as H-based than Vorono -based shapes.
Tentative explanations for that are proposed in next section.
5 Epistasis and Evolutionary Schemes
5.1 Epistasis and Symmetry of Representa- tions
In the biological context, the epistasis is a measure of how the expression in the phenotype of one single gene is inuenced by the other genes of the genotype. In Evo- lutionary computation, epistasis has a strong inuence on the way genetic material is transmitted from parents to ospring, and how it can be modied by evolution operators thereafter applied.
In the bit-array representation, the contribution of a given bit to the phenotype is clear, and it cannot modied by any other bit. The epistasis is here mini- mal, all genetic material transmitted to the ospring is expressed, i.e. is dominant. Such a situation can be termed strong transmission.
In the Vorono representation, the inuence of a site on the nal phenotype can be greatly modied by other neighbor sites. If one site labeled void becomes surrounded by sites labeled material, its inuence on the nal phenotype almost vanishes. Some part of genetic information transmitted to ospring is reces- sive, and, by contrast to the bit-array situation, this case will be termed weak transmission.
In the H-representation, the situation is even more complex: Only holes are strongly transmitted. When a hole is transmitted from parent to ospring, the whole area will stay a hole. On the opposite, no such strong transmission is achieved for \non-holes". The transmission is asymmetric, strong for the hole value, and weak for the default value (i.e. any hole covering the same area changes its value). One can say the hole-value represent dominant genetic material, while the default value is recessive.
The bottom-up approach of GA, trying to recombine building blocks to reach the optimal solution should be much more hindered by epistasis than the top-down ap- proach only relying on the competition between pheno- types to reach an optimal point of the search space. In order to check these points, experiments were conducted on the benchmark cantilever problem (section 2.3) using the two dierent approaches of EC.
5.2 Evolutionary Schemes
Evolutionary Algorithms (EA) crudely mimic the evolu- tion of a population of points of the search space. This population is usually initialized randomly, and under- goes a succession of generations. The general outline of a generation can be viewed as
SELECT
parents from the populationAPPLY
evolutionary operators to the selected parents to generate offspringREPLACE
some parents by some offspringThe most widely used EAs, namely Evolutionary Pro- gramming [Fogel, Owens and Walsh 1966], Evolution- ary Strategies [Schwefel 1981] and Genetic Algorithms [Holland 1975], are instances of this general scheme:
EP and ES do not use initial selection process while GA uses tness-based stochastic selection (e.g.
roulette wheel).
EP uses mutation only, each parent generating one ospring, ES uses mostly mutation, each parent gen- erating usually more than one ospring and GA use mostly recombination (crossover), two parents gener- ating two ospring.
EP replaces the parents using a stochastic tourna- ment among parents and ospring, ES selects the best individuals among parents and ospring in (+)
ESor among ospring only in (;) ESas the new parents, and GA globally replaces all parents with all ospring.
Of course, numerous variations of these canonical algo- rithms exist and are being used in practical applications on a pragmatic basis [Michalewicz 1992, Fogel 1995a, Back 1995]. However, the a priori adequation of an evo- lutionary scheme to a given application is still an open question.
5.3 Experimental results
Experiments were conducted using two evolutionary schemes:
The standard generational GA, with population size of 100, using ranked-based selection, crossover rate of 0.6, mutation rate per individual of 0.2, with Gaus- sian mutations of xed variances.
A (15+100)-ES (15 parents generate 100 ospring, the best 15 among parents + ospring become the parents of the next generation) with strength of Gaussian mutations depending on the tness of the individual at hand as in EP [Fogel 1992].
In both cases, the maximum number of genera- tions allowed is set to 100, and the algorithm stops if no improvement is observed during 10 genera- tions. These stopping criterions are severe, and most runs did not reach convergence. But, in contrast with [Kane, Jouve and Schoenauer 1995, Kane 1996], the goal here is to observe the behavior of the algorithms with the perspective of fast convergence, rather than to reach the optimal shape by all means.
As one of the goals is to study the impact of the symmetry of the representations on the evolutionary algorithm, three representations are considered: the Vorono representation (section 4.1), the \Hole" repre- sentation and the \Plate" representation, which are both H-representation (section4.2), where the default value is material in the case of the Hole-representation, and void in the case of the Plate-representation (the structure is here an assembly of small plates).
In order to test the inuence of both the epistasis and the symmetry of the representation interacting with the
Generations
Fitness
0 20 40 60 80
0.5 0.6 0.7
Holes - ES Holes - GA Plates - ES Plates - GA Vor. - ES Vor. - GA Max. Dep. 20, Alpha 1
Figure 4: .Comparative results between combinations of representations and evolutionary schemes: Averaged (over 30 independent runs) best tness along genera- tions. The constraint on the displacement is very strong (limit value 0.2) and the penalty parameter is small.
evolution scheme, dierent instances of the cantilever plate problem described in section 2.3 are considered.
For all problems, the design domain is the 21 rectan- gle, the mesh is a 2010 regular mesh (i.e. all elements are here squares) and the same force is applied at point (2;0:5) (pointM in Figure 1).
The maximum value of the displacement is assigned dierent values, increasing from 0.2, the actual displace- ments of the full horizontal beam joining point M and the xed boundary of thickness 0:8 to 37.10, the actual displacement of the full beam of thickness 0:1. Note that only in this latter case is the optimum known: due to the coarse mesh used here, the minimal structure connect- ing pointM and the xed boundary is the full beam of thickness 0:1, which does indeed respect the constraint.
Finally, the penalty parameter is varied too: intu- itively, the higher the penalty factor, the more di- cult the problem (feasible regions are environed by steep slopes).
It is fair to say that only tendencies could be observed, and no absolute conclusion can be drawn.
1. Both H-representations globally outperform the Vorono representation. So the high degree of epista- sis seems to penalize the representation here.
2. ES and GA are hardly distinguishable, except { when the limit value for the displacement is small (strong constraint), in which case GA signicantly outperforms ES for all representations (see Figure 4). However, this phenomenon decreases when the penalty parameter increases. A possible explanation could be that the elitist ES gets stuck more easily in the rst feasible local optimum for small values of the penalty parameter, whereas both algorithms encounter this same diculty for large values of the penalty parameter.
Generations
Fitness
0 20 40 60 80
0.2 0.3
0.25
0.35 Holes - ES
Holes - GA Plates - ES Plates - GA Vor. - ES Vor. - GA Max. Dep. 640, Alpha 20
Figure 5: Comparative results between combinations of representations and evolutionary schemes: Averaged (over 30 independent runs) best tness along genera- tions. The constraint on the displacement is weak (limit value 6.40) and the penalty parameter is large.
{ for the Vorono representation with large limit for the displacement (weak constraint) in which case ES performs better than GA (see Figure 5). This is the only hint meeting the a priori expectation that epista- sis should favor the top-down against the bottom-up approach.
3. The \Hole" representation demonstrates slightly bet- ter performances than the \Plate" representation in almost all cases (e.g. in Figure 5) except when the optimal solution contains about as much material as void, as can be seen in Figure 4: a value of 0:5 for the tness of feasible structures corresponds to ex- actly the same amount of void and material.
6 Further directions for the comparison of representations
Overall, these experiments show that far too many pa- rameters are involved in the TOD problem to make it a good test-bed for shape representation ! A next step will be to design such an adequate test-bed, as an un- constrained problem the solution of which is known, with tunable amount of void and material, and scal- able degree of diculty. On-going work is concerned with the problem of (non-destructive) identication of the repartition of two materials in a structure (e.g.
scories in a steel piece) using mechanical experiments [Constantinescu 1994].
Some criterions investigated in the literature will guide systematic experiments:
The tness variance theory of Rad-
clie [Radclie and Surry 1994] studies the variance of the tness as a function of the order of an extension of schemas called formae [Radclie 1991], and, sim- ply put, shows that the complexity and diculties of evolution increases with the average variance of the
tness. But if the formae and their order (or their precision) are well-dened on any binary representa- tion, including the bit-array representation of section 3, it is not straightforward to extend these denitions to variable length representations discussed above.
Moreover, Radclie's tness variance does not take into account the possible evolution operators. Fur- ther step in that direction would be to study the vari- ance of the change of tness with respect to a given evolution operator (e.g. the Gaussian mutation of Vorono sites for dierent standard deviations), as in [Fogel 1995b]
The tness distance correlation of Jones and Forrest [Jones and Forrest 1995] studies the correlation be- tween the distance to the optimal point and the t- ness. Simply put again, the idea is: the stronger this correlation, the narrower the peak the optimum belongs to, and the more dicult the problem. Con- jectures based on this remark are experimentally con- rmed in the GA-frame. Nevertheless, the diculty in shape representation is to dene a distance which is meaningful for both the representation and the prob- lem at hand. The rst important issue is whether the considered distance should be purely genotype-based (i.e. dened on the coded individual only) or par- tially or totally phenotype-based (i.e. dened on the same space than the tness function).
7 Conclusion and further work
Dierent evolutionary approaches for shape optimization have been presented. The emphasis has been put on the representation: the simple bit-array representation allowed signicant advances in the domain of topological shape optimization (e.g. the rst results in nonlinear elasticity), but hardly scales up, in the sense that the accuracy of the tness computation is commanded by the size of the individuals.
Two other, variable-length, representations, the Vorono and the H- representations have been designed to overcome this issue; they demonstrate good results on the Topological Optimum Design problem, signicantly and consistently outperforming the bitarray representa- tion.
The main dierence between those latter representa- tions a priori appears their respective degree of epistasis, i.e. the way one gene (Vorono site or hole) may mod- ify the phenotypic traits due to other genes. System- atic experiments have been conducted to see how the de- gree of epistasis interacts with the evolutionary scheme (bottom-up as in GAs or top-down as in ES) and the symmetry of the representation ... and mainly demon- strate that the classical benchmark in Topological Opti- mum Design is a problem far too complex to serve as a test-bed for evaluating shape representations.
A number of powerful representations of shapes remain to be investigated in the frame of evolutionary optimiza- tion:
The CAD community uses splines dened from con- trol points to describe and manipulate shapes.
The L-system paradigm
[Prusinkiewicz and Lindenmeyer 1990] uses gram- mar rules to simulate plant growth. The result- ing \plants" can be viewed as shapes. Moreover, it has been demonstrated that such grammar rules can be optimized using Evolutionary Computation algo- rithms [Simms 1994].
The fractal theory of Iterated Function Systems [Barnsley 1988] oers another possible representa- tion, for which the inverse problem has been success- fully tackled by means of Evolutionary Computation [Garigliano & al. 1993, Lutton and Martinez 1994].
It is emphasized however, that a key problem of evolu- tionary computation yet is the choice of an adequate rep- resentation. Widening this choice would only ask more loudly for judicious choice criterions; and it is our convic- tion that the rst, and maybe the more important step on this way, would be to elaborate a convenient test-bed, of tunable diculty.
On-going work considers another shape identica- tion problem to this end, namely the identica- tion of inclusions (e.g. scories in a piece of steel) [Constantinescu 1994]. This latter problem presents two main advantages: it is unconstrained and fully tunable (the optimal solution can be xed a priori).
Acknowledgements
Grateful thoughts to Michele Sebag for useful discussions and helpful advices.
References
[Allaire & al. 1996] Allaire, G., E. Bonnetier, G. Francfort, and F. Jouve (1996). Shape optimization by the homoge- nization method.To appear in Numerische Mathematik. [Allaire and Kohn 1993] Allaire, G. and R. V. Kohn (1993).
Optimal design for minimum weight and compliance in plane stress using extremal microstructures. European Journal of Mechanics, A/Solids, 12(6):839{878.
[Back 1995] Back, T. 1995). Evolutionary Algorithms in the- ory and practice. New-York:Oxford University Press.
[Barnsley 1988] Barnsley, M. F. (1988).Fractals everywhere. Academic Press.
[Bendsoe and Kikushi 1988] Bendsoe, M. and N. Kikushi (1988). Generating optimal topologies in structural de- sign using a homogenization method.Computer Methods in Applied Mechanics and Engineering, 71:197{224.
[Boissonnat and M. Yvinec 1995] Boissonnat, J.-D. and M. Yvinec (1995). Geometrie algorithmique. Ediscience International.
[Caruna and Schaer 1988] Caruna, R. A. and J. D. Schaf- fer (1988). Representation and hidden bias : Gray vs binary coding for genetic algorithms. InProceedings of ICML-88, International Conference on Machine Learn- ing. Morgan Kaufmann.
[Cea 1981] Cea, J. (1981). Problems of shape optimum de- sign. In E. J. Haug and J. Cea, editors, Optimization of distributed parameter structures - Vol. II, volume 50, pages 1005{1088. NATO Series, Series E.
[Cerf 1994] Cerf, R. (1994). Une theorie assymptotique des algorithmes genetiques. PhD thesis, Universite de Mont- pellier II, March 1994.
[Cerf 1996] Cerf, R. (1996). An asymptotic theory of ge- netic algorithms. In J.-M. Alliot, E. Lutton, E. Ronald, M. Schoenauer, and D. Snyers, editors,Articial Evolu- tion { EA95, LNCS. Springer Verlag.
[Chapman and Jakiela 1995] Chapman, C. D. and M. J.
Jakiela (1995). Genetic algorithm-based structural topology design with compliance and topology simpli- cation considerations. Journal of Mechanical Design. Submitted.
[ Chapman, Saitou and Jakiela 1994] Chapman, C.
D., K. Saitou, and M. J. Jakiela (1994). Genetic al- gorithms as an approach to conguration and topology design. Journal of Mechanical Design, 116:1005{1012.
[Ciarlet 1978] Ciarlet, P. G. (1978). Mathematical Elastic- ity, Vol I : Three-Dimensional Elasticity. North-Holland, Amsterda.
[Ciarlet 1988] Ciarlet, P. G. (1988). The Finite Element Method for Elliptic Problems. North-Holland, Amster- [Constantinescu 1994] Constantinescu, A. (1994).dam. Sur l'identication des modules elastiques. PhD thesis, Ecole Polytechnique.
[Dejonghe 1993] Dejonghe, J., (1993). Allegement de platines metalliques par algorithmes genetiques. Rapport de stage d'option B2 de l'Ecole Polytechnique.
[Fogel 1992] Fogel, D. B. (1992). An analysis of evolutionary programming. In D. B. Fogel and W. Atmar, editors, Proceedings of the 1st Annual Conference on Evolution- ary Programming, pages 43{51. Evolutionary Program- ming Society.
[Fogel 1995a] Fogel, D. B. (1995). Evolutionary Computa- tion. Toward a New Philosophy of Machine Intelligence. IEEE Press.
[Fogel 1995b] Fogel, D. B. (1995). Phenotypes, genotypes and operators in evolutionary computation. In D. B.
Fogel, editor,Second IEEE International Conference on Evolutionary Computation. IEEE Press.
[Fogel, Owens and Walsh 1966] Fogel, L. J., A. J. Owens and M. J. Walsh (1966). Articial Intelligence through Sim- ulated Evolution. New York: John Wiley.
[Garigliano & al. 1993] Garigliano,R. A. Purvis, P. A. Giles and D. J. Nettleton (1993). Genetic algorithms and shape representation. In D. B. Fogel and W. Atmar, editors, Proceedings of the 2nd Annual Conference on Evolutionary Programming, pages 40{47. Evolutionary Programming Society.
[George 1991] George, P. L. (1991). Automatic mesh gener- ation, application to Finite Element Methods. Wiley &
Sons.
[Ghaddar, Y. Maday, and A. T. Patera 1996] Ghaddar, C., Y. Maday and A. T. Patera 1996). Analysis of a part design procedure. Submitted to Numerishe Mathematik. [Goldberg and Richardson 1987] Goldberg, D. E. and J. Richardson (1987). Genetic algorithms with sharing for multi-modal function optimization. In J. J. Grefen- stette, editor,Proceedings of the2ndInternational Con- ference on Genetic Algorithms, pages 41{49. Lawrence Erlbaum Associates.
[Holland 1975] Holland, J. (1975).Adaptation in natural and articial systems. University of Michigan Press, Ann Arbor, 1975.
[Jensen 1992] Jensen, J. (1992). Topological Structural De- sign using Genetic Algorithms. PhD thesis, Purdue Uni- versity.
[Jones and Forrest 1995] Jones, T. and S. Forrest (1995).
Fitness distance correlation as a measure of problem dif- culty for genetic algorithms. In L. J. Eshelman, editor, Proceedings of the6th International Conference on Ge- netic Algorithms, pages 184{192. Morgan Kaufmann.
[Kahng and Moon 1995] Kahng, A. B. and B. R. Moon (1995). Toward more powerful recombinations. In L. J.
Eshelman, editor, Proceedings of the 6th International Conference on Genetic Algorithms, pages 96{103. Mor- gan Kaufmann.
[Kane 1996] Kane, C. (1996). Algorithmes genetiques et Op- timisation topologique. PhD thesis, Universite de Paris VI.
[Kane, Jouve and Schoenauer 1995] Kane, C., F. Jouve and M. Schoenauer (1995). Structural topology optimiza- tion in linear and nonlinear elasticity using genetic al- gorithms. InProceedings of the ASME 21st Design Au- tomation Conference. ASME.
[Kane and Schoenauer 1995] C. Kane, C. and M. Schoenauer (1995). Genetic operators for two-dimensional shape optimization. In J.-M. Alliot, E. Lutton, E. Ronald, M. Schoenauer, and D. Snyers, editors,Articial Evolu- tion { EA95. Springer Verlag. To appear.
[Lutton and Martinez 1994] Lutton, E. and P. Martinez (1994). A genetic algorithm with sharing for the detec- tion of 2d primitives in images. In J.-M. Alliot, E. Lut- ton, E. Ronald, and M. Schoenauer, editors,Actes de la Conference Evolution Articielle. Cepadues, Toulouse.
[Michalewicz 1992] Michalewicz, Z. (1992). Genetic Algo- rithms+Data Structures=Evolution Programs. Springer Verlag.
[Michalewicz 1995] Michalewicz, Z. (1995). A survey of con- straint handling techniques in evolutionary computation methods. In J. R. McDonnell, R. G. Reynolds, and D. B.
Fogel, editors,Proceedings of the 4thAnnual Conference on Evolutionary Programming.
[Preparata and Shamos 1985] Preparata, F. P. and M. I.
Shamos (1985). Computational Geometry: an introduc- tion. Springer Verlag.
[Prusinkiewicz and Lindenmeyer 1990] Prusinkiewicz, P.
and A. Lindenmeyer (1990). The Algorithmic Beauty of Plants. Springer Verlag, New York.
[Radclie 1991] Radclie, N. J. (1991). Equivalence class analysis of genetic algorithms.Complex Systems, 5:183{
[Radclie and Surry 1994] Radclie, N. J. and P. D. Surry.20.
Fitness variance of formae and performance prediction.
In D. Whitley and M. Vose, editors,Foundations of Ge- netic Algorithms 3, pages 51{72. Morgan Kaufmann.
[Schoenauer 1995] Schoenauer, M (1995). Representations for evolutionary optimization and identication in struc- tural mechanics. In J. Periaux and G. Winter, editors, Genetic Algorithms in Engineering and Computer Sci- ences. John Wiley.
[[Schoenauer and Xanthakis 1993] Schoenauer, M.
and S. Xanthakis (1993). Constrained GA optimization.
In S. Forrest, editor,Proceedings of the5thInternational Conference on Genetic Algorithms, pages 573{580. Mor- gan Kaufmann.
[Schwefel 1981] Schwefel, H.-P. (1981). Numerical Optimiza- tion of Computer Models. John Wiley & Sons, New- York.
[Seguin 1995] Seguin, M. (1995). Optimisation de formes par evolution articielle. etude de deux representations.
Rapport de DEA d'Analyse Numerique de l'Universite de Paris VI.
[Simms 1994] Simms, K. (1994). Evolving virtual creatures.
InSIGGRAPH'94 Proceedings, pages 15{22. Computer Graphics, Annual Conference Series, July 1994.
[Zienkiewicz 1977] Zienkiewicz, O. C. (1977). The Finite El- ement Method in Engineering Science. McGraw-Hill, New-York, 3rd edition.
